1. Is there an analogue of the Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup?
2. What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
3. What is the best technique to get the spectral sequence? For me the Grothendieck spectral sequence us much better than the spectral sequence of a filtered complex.

There is a parallel question which is likely easier.

1. Is there an analogue of the Hochschild–Serre spectral sequence for a Lie subalgebra which is not an ideal?

2 and 3 remain the same.

I already asked a version of this question on MathSE but got no responses.